Stability and bifurcation analysis of a discrete predator\u2013prey system with modified Holling\u2013Tanner functional response

Jianglin Zhao,Yong Yan

doi:10.1186/s13662-018-1819-0

Abstract

In this paper, we study a discrete predator–prey system with modified Holling–Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.

Highlights

Predator–prey interactions have long been studied and continue to be one of the dominant themes in both biology and mathematical biology due to their universal existence and importance [1]
5 Conclusion In this paper, we have mainly considered the complex behaviors of a predator–prey system with modified Holling–Tanner functional response in R2
By using the center manifold theorem and bifurcation theory we prove that the unique positive fixed point of system (3) can undergo flip bifurcation and Hopf bifurcation

Summary

Introduction

Predator–prey interactions have long been studied and continue to be one of the dominant themes in both biology and mathematical biology due to their universal existence and importance [1]. The characteristic equation of the Jacobian matrix J(u, v) evaluated at the unique positive fixed point B(u∗, v∗) can be written as λ2 – (2 + Gδ)λ + 1 + Gδ + Hsδ2 = 0,. From the Jury criterion and the preceding analysis it can be seen that one of the eigenvalues of the unique positive fixed point B(u∗, v∗) is –1 and the other is neither 1 nor –1 if (iv.1) of Theorem 2.3 holds. The unique positive fixed point B(u∗, v∗) may undergo the flip bifurcation when parameters vary in a small neighborhood of FB1 or FB2. The unique positive fixed point B(u∗, v∗) may undergo the Hopf bifurcation when the parameters vary in a small neighborhood of HB

Flip bifurcation and Hopf bifurcation

Conclusion